Eight possible magnetic states \({\rm S}_1\)  to \({\rm S}_8\)  are shown in Fig. 1(c). In Table I, we summarize the proximity-induced magnetism in graphene materials reported in the existing literature [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}. It is seen that there have been reports of direct relevance for S\(_1\)  to S\(_4\) . For S\(_5\)  to S\(_8\) , no direct theoretical or experimental reports have been found yet, however we provide information on the single-interface proximity case, i.e. monolayer graphene/magnetic insulator, as a reference. The configurations S\(_5\)  to S\(_8\)  become available if we select different magnets on the top and bottom sides. Taking S\(_7\)  and S\(_8\)  for example, they can be realized by contacting BLG with a ferromagnetic insulator from the bottom side and with an antiferromagnetic insulator from the top side. To study the electronic properties of the heterostructures, we use a simple assumption that the proximity effect only induces magnetism in BLG but does not change the band features of BLG, as proven previously [2]}, [3]}, [4]}, [5]}, [6]}, [8]}, [9]}, [10]}, [11]}, [19]}, [20]}. Specifically, the applied vertical pressure in heterostructures [16]}, [18]} is helpful to enhance the proximity-induced magnetism and avoid the introduction of extra orbital hybridization near the Fermi energy of graphene materials. Note that, since the magnets that induce magnetism from the top or bottom side of BLG basically only affect a single layer of BLG close to it, with another graphene layer being largely unaffected [6]}, [3]}, [5]}, [4]}, [20]} under the ultra-short range proximity effect, the patterns \({\rm S}_1\)  to \({\rm S}_8\)  for the Bernal-stacking situation we consider, the most stable structure and mostly realized in experiments [39]}, [40]}, [41]}, are also applicable to the relatively-rare AA-stacking situation [42]}, [43]}. In the atomic basis \(\lbrace \psi _{A_1\uparrow }\) , \(\psi _{A_1\downarrow }\) , \(\psi _{B_1\uparrow }\) ,
\(\psi _{B_1\downarrow }\) , \(\psi _{A_2\uparrow }\) , \(\psi _{A_2\downarrow }\) , \(\psi _{B_2\uparrow }\) ,
\(\psi _{B_2\downarrow }\rbrace \) , we establish a generalized phenomenological Hamiltonian as follows,
\(\begin{split}{\cal H}=&{\mathbf {I}_\tau }\otimes \upsilon (\sigma _xp_x+\xi \sigma _yp_y)\otimes {\mathbf {I}_s}\\&+\frac{\gamma }{2}(\tau _x\otimes \sigma _x-\tau _y\otimes \sigma _y)\otimes {\mathbf {I}_s}\\&+M\Sigma _{\tau ,\sigma }\otimes s_z\\&+\Delta \tau _{z,\eta ^{\prime }}\otimes \sigma _z\otimes {\mathbf {I}_s}\\&+U\tau _z\otimes {\mathbf {I}_\sigma }\otimes {\mathbf {I}_s}\\&+{\cal H}^{\prime },\\{\cal H}^{\prime }=&\frac{\lambda _R}{2}\tau _{z,\nu }\otimes (\sigma _x\otimes s_y-\xi \sigma _y\otimes s_x)\\&+\lambda _I\xi \tau _{z,\chi }\otimes {\mathbf {I}_\sigma }\otimes s_z.\end{split}\) 
